In this question, I would love to hear some independent opinions on an issue I asked Juan Maldacena, Nathan Berkovits, Dan Jafferis, and others, but all the physicists may be missing something. The question has 2 main parts:

  1. Is there a dual superconformal symmetry in 2+1-dimensional ABJM theories? In what limit does it apply? (For $N=4$ gauge theory, it's the planar limit, but what is the counterpart in 2+1 dimensions?)
  2. Are there spin network states in this extension of Chern-Simons theory? Can their expectation values be related to scattering amplitudes in the ABJM theory, much like the piecewise null Wilson loops are related to scattering amplitudes in $N=4$ gauge theory?

Concerning the first question, there's some confusion what $N$ should go to infinity and what should be kept fixed, especially when it comes to the scaling of the level $k$ in the limit. In string theory, the dual superconformal symmetry (and the Yangian) is very constraining - but that's because it still constrains a lot of the "stringy stuff".

Should one necessarily go to the limit with the stringy stuff - e.g. a high-level, a $CP^3$ compactification of type IIA - to see some dual superconformal symmetry? Or is there a dual superconformal symmetry that only works in the planar limit which simply means that only the (leading?) SUGRA amplitudes are constrained?

Concerning the second question, the ABJM theory secretly contains membranes, and their discretization - analogous to the piecewise null Wilson loops - could involve faces (pieces of membranes?) surrounded by open Wilson lines. Are they gauge-invariant?

  • $\begingroup$ What is the ABJM theory? ... Ok. I got it. Aharony, Bergman, Jafferis and Maldacena $\endgroup$ – user346 Feb 20 '11 at 9:36
  • $\begingroup$ can you point me to sources where I can learn how piecewise null Wilson loops are related to scattering amplitudes in N=4 gauge theory? $\endgroup$ – user346 Feb 20 '11 at 10:12
  • $\begingroup$ Yes, ABJM is them: arxiv.org/abs/0806.1218 $\endgroup$ – Luboš Motl Feb 21 '11 at 15:26
  • $\begingroup$ The piecewise null Wilson loops to calculate the amplitudes were started by Alday and Maldacena etc. xxx.lanl.gov/abs/0705.0303 but see my popular review of the twistor uprising for other references: motls.blogspot.com/2011/01/twistor-minirevolution-goes-on.html $\endgroup$ – Luboš Motl Feb 21 '11 at 15:31

My wild idea would be to look for holomorphic linking of Wilson loops in the ABJM theory, working from its conjectured Grassmannian integral representation.

  • $\begingroup$ Very interesting... $\endgroup$ – Luboš Motl Feb 21 '11 at 15:25

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